We report the detection of optical and X-ray variability in the least luminous known Seyfert galaxy, NGC 4395. Between 1996 July and 1997 January the featureless continuum changed by a factor of 2, which is typical of more luminous AGN. The largest variation was seen at shorter wavelengths, so that the spectrum became `harder' during higher activity states. During the same period, the broad emission-line flux changed by ∼ 20-30 per cent. In a one-week optical broad-band monitoring programme, a 20 per cent change was seen between successive nights. The difference in flux observed between the spectroscopy of 1996 July and the broad-band observations implies variation by a factor of 3 at 4400 A in just one month. In the same period, the spectral shape changed from a power law with spectral index α∼ 0 (characteristic of quasars) to a spectral index α∼ 2 (as observed in other dwarf AGN). ROSAT HRI and PSPC archive data show a variable X-ray source coincident with the Galactic nucleus. A change in X-ray flux by a factor ∼ 2 in 15 d has been observed. When compared with more luminous AGN, NGC 4395 appears to be very X-ray quiet. The hardness ratio obtained from the PSPC data suggests that the spectrum could be absorbed. We also report the discovery of weak Ca II K absorption, suggesting the presence of a young stellar cluster providing of the order of 10 per cent of the blue light. The stellar cluster may be directly observed as a diffuse component in Hubble Space Telescope (HST ) optical imaging. Using HST UV archive data, together with the optical and X-ray observations, we examine the spectral energy distribution for NGC 4395 and discuss the physical conditions implied by the nuclear activity under the standard AGN model. When in the low state, the extrapolated UV continuum is insufficient to explain the observed broad emission lines. This could be explained by intrinsic variability or absorption or may imply an extra heating source for the broad-line region. The observations can be explained either by an accreting massive black hole emitting at about 10-3LEdd or by a single old compact supernova remnant with an age of 50 to 500 yr generated by a small nuclear starburst.